In the field of satellite navigation, a position of an object, which can be a stationary object or a moving object, is determined using radio signals transmitted by satellites orbiting earth. In the case of a moving object, information relating to the movement of the object, for instance, its direction and speed, can be obtained in this manner. Commonly, in the field of satellite navigation, the object is referred to as a “rover,” or also as a “user.” Without intended limitation, in the following, it will be referred to a “rover” or a “user” when describing the present disclosure.
There are various techniques known in the art for determining the position of a rover using satellite-transmitted radio signals. All of these techniques require the knowledge of the instantaneous position of a number of satellites orbiting earth (wherein the satellites may also be geostationary satellites) and the knowledge of high-precision clocks installed aboard these satellites. The satellite-transmitted radio signals comprise time-stamped information that is transmitted by superimposing a pseudo-random signal on the carrier frequencies of the signals. Upon reception, so-called code observables or pseudorange observables are determined from the time-stamped information. Having knowledge of the position of the satellites, the time of transmission of the signals, and the time of reception of the signals at the rover, the coordinates of the rover with respect to the satellites may be determined by triangulation.
Apart from the above-described time-stamped information that is superimposed on the carrier frequency, the carrier phase can be used to determine the position of a rover. Based on the phases of the carrier signals, so-called phase observables can be obtained and used in determining the position of the rover, as will be described below.
A widely employed satellite navigation system is the “Global Positioning System,” or “GPS.” GPS satellites transmit signals into two frequency bands, generally referred to as L1 with a carrier frequency of 1.575 GHz and L2 with a carrier frequency of 1.227 GHz. Thus, the GPS is referred to as a “dual frequency” system. Future satellite navigation systems are the “Galileo” system and the “modernized GPS” system, which will provide satellites transmitting in three frequency bands. In the following, these (and comparable) systems will be summarizingly referred to as “Global Navigation Satellite Systems,” or “GNSS.”
A known technique for determining the position of a rover using signals of three different carrier frequencies is the “Wide Area Real Time Kinematic,” or “WARTK” technique, as described in U.S. Pre-Grant Publication No. 2006/0164297. According to the WARTK technique, the rover needs to be in contact with a network of fixed reference stations also receiving the satellite-transmitted signals. Herein, being in contact with the network of reference stations is meant such that the rover receives data relating to radio signals received at one or more of the reference stations of the network from the network via a ground-based or space-based means of communication. Both the rover and the reference stations obtain pseudorange observables and carrier-phase observables from the satellite-transmitted signals, and the network provides, via a ground-based or space-based means of communication, the pseudorange observables and carrier-phase observables obtained at the one or more reference stations to the rover. In the rover, double differences between satellite-receiver pairs of the observables are taken. If, for instance, O11 is an observable relating to a signal of a first satellite obtained at the rover, O12 is an observable relating to a signal from a second satellite received at the rover, O21 is an observable relating to a signal from the first satellite obtained at the reference station and O22 is an observable relating to a signal from the second satellite obtained at the reference station, such a double difference would be given by (O11−O12)−(O21−O22). A more detailed description of double-differences is given in the article “Analysis of Three Carrier Ambiguity Resolution (TCAR) Technique for Precise Relative Positioning in GNSS-2,” by U. Vollath et al., published in “Proceedings of the ION GPS,” 1988, ix-o-13, pages 1-6.
By taking these double differences, a number of observation errors and instrumental errors are canceled. However, the maximal distance between the rover and the reference stations is limited, because the signals received by the rover and the signals received by the reference stations are affected differently by delays occurring due to the passage of these signals through the ionosphere. Therefore, with increasing distance between the rover and the reference stations, the observables obtained at the rover and at the reference stations are de-correlated by the ionospheric delays. This problem is addressed by a real-time ionospheric model which is computed from the network of reference stations and provided to the rover via a ground-based or space-based means of communication. This model is determined from an analysis of dual-frequency carrier-phase and code observables obtained at the reference stations of the network. By way of this ionospheric model, the effect of the de-correlation of observables obtained at the rover and at the reference stations may be determined and taken into account in determining the rover position. Thereby, the impact of the above-described de-correlation may be mitigated.
While with this approach, possible separations between the rover and the reference stations of several hundred kilometers can be obtained, the network of reference stations required for performing the WARTK technique still needs to be relatively dense. On the other hand, the transmission of the observables from the reference receivers to the rover receiver requires a high bandwidth of the transmission channel, and this requirement limits considerably the number of possible reference receivers. Therefore, coverage for a navigation service employing the WARTK technique is feasible at most on a continental scale, but providing a world-wide navigation service based on the WARTK technique in view of this limitation is not possible.
In view of the above explanations, it also becomes evident that the WARTK technique is only applicable in regions in which reference stations are provided. In a region without reference stations, or at a position remote from reference stations, the determination of the rover position is considerably impeded, if not made impossible. Therefore, the WARTK technique is typically not deployable for navigation in large areas of wilderness, in mountain ranges, or generally in impassable areas, or for navigation on large bodies of water or deserts. Accordingly, an application of the WARTK technique is only of very limited use for aviation, seafaring, or exploration. Furthermore, breakdown of individual reference stations, as might occur due to local events such as severe weather conditions or electrical power outage, considerably impede navigation of rovers in an area in which a broken-down reference station is the nearest reference station.
An alternative technique for determining the position of a rover is the “Precise Point Positioning,” or “PPP” technique, as discussed for instance in J. F. Zumberge et al., “Precise point positioning for the efficient and robust analysis of GPS data from large networks,” Journal of Geophysical Research, Vol. 102, No. B3, pp. 5005-5017, 1997 doi:10.1029/96JB03860. PPP is considered as a technique that allows a multiple-frequency GNSS user to determine his position at the decimeter error level in a kinematic mode and at the centimeter level in a static mode with a single receiver. This is based on the availability of satellite products or service data, such as data relating to the orbits of the GNSS satellites and data relating to the internal clocks of the GNSS satellites (i.e., the difference between the satellite clocks with respect to the GNSS time scale, hereinafter referred to as satellite clocks). For the purposes of the PPP technique, these satellite products need to be significantly more precise than those computed, e.g., by the GPS control segment. Therefore in the context of the PPP technique, the data relating to the orbits of the GNSS satellites is commonly referred to as “precise orbits,” and the data relating to the internal clocks of the GNSS satellites is commonly referred to as “precise clocks.”
The basic features of the PPP technique will be described in the following with reference to FIG. 10.
In step S1201, precise clocks and precise orbits of the GNSS satellites are received by the user. These precise clocks and precise orbits are provided by a GNSS service provider. Assuming that at a given timing a number n of GNSS satellites are in view of the user, the positions of these satellites predicted from the precise orbits are denoted {right arrow over (r)}i and the internal clocks of these GNSS satellites are denoted dti for i=1, . . . , n denoting a particular GNSS satellite. Here and in the following, “in view of the user” is meant to be understood in a sense that a signal transmitted by a GNSS transmitter in view of the user can travel to a GNSS receiver associated to the user substantially on a straight line (line of sight between the GNSS transmitter and the GNSS receiver) without being obstructed by solid objects such as, for example, buildings, mountains, or the earth itself. In the following, data provided by such a service provider, comprising, but not being limited to, precise clocks and precise orbits, will be referred to as “service data” 1206.
In step S1202, signals 1207 of two different carrier frequencies transmitted by GNSS transmitters aboard the satellites in view of the user are received by a GNSS receiver associated to the user. If the two different carrier frequencies are denoted by f1 and f2, n signals of carrier frequency f1 and n signals of carrier frequency f2 are received by the user (by the GNSS receiver associated to the user).
At step S1203, n carrier-phase observables L1i relating to the carrier phases of the signals of carrier frequency f1; n carrier-phase observables L2i relating to the carrier phases of the signals of carrier frequency f2; n code observables P1i relating to the time-stamped information transmitted with the signals of carrier frequency f1; and n code observables P2i relating to the time-stamped information transmitted with the signals of carrier frequency f2 are obtained from these signals.
A code observable Pji relating to a signal of carrier frequency fj is derived from a time difference between signal reception at the GNSS receiver and a time of signal transmission at the corresponding GNSS transmitter via Pji=c(tk−ti), where tk is the reception time, measured by the receiver clock, ti is the transmission time, measured by the GNSS transmitter clock, and c is the vacuum speed of light. Said time difference can be obtained from time-stamped information constituted by a pseudo-random code that is superimposed on the carrier wave of the signal. A carrier-phase observable Lji relating to a signal of carrier frequency fj is derived from a difference between a phase of the carrier wave of the signal at reception time and a phase of the carrier signal at transmission time via Lji=λj(φk−φi)+λjNji, where φk is the phase of the carrier signal at reception time, including any receiver carrier-phase bias, φi is the phase of the carrier phase at transmission time, including any transmitter carrier-phase bias, λj is the wavelength of the carrier signal, and Nji is an integer number of full cycles. The integer number of full cycles Nji unknown since only a fractional phase is measured. All observables are affected by a number of instrumental and/or observational errors discussed below.
In step S1204, so-called ionospheric-free carrier-phase observables Lci and ionospheric-free code observables Pci are determined from the carrier-phase observables L1i, L2i and the code observables P1i, P2i by forming linear combinations of these observables. Specifically, the ionospheric-free combinations Lci and Pci are given by:
                              L          c          i                =                                                                              (                                      f                    1                                    )                                2                            ⁢                              L                1                i                                      -                                                            (                                      f                    2                                    )                                2                            ⁢                              L                2                i                                                                                        (                                  f                  1                                )                            2                        -                                          (                                  f                  2                                )                            2                                                          (                  Eq          .                                          ⁢          1                )                                          P          c          i                =                                                                              (                                      f                    1                                    )                                2                            ⁢                              P                1                i                                      -                                                            (                                      f                    2                                    )                                2                            ⁢                              P                2                i                                                                                        (                                  f                  1                                )                            2                        -                                          (                                  f                  2                                )                            2                                                          (                  Eq          .                                          ⁢          2                )            
Since the ionospheric delay of a signal of carrier frequency f experienced when traveling through the ionosphere is proportional to 1/f2, the effects of ionospheric delays of the signals of carrier frequencies f1 and f2 cancel in the above combinations Lci and Pci to good precision. In practice, a cancellation of 99.9% of the effects of ionospheric delays of the signals can be achieved.
If the present position of the rover k is denoted by {right arrow over (r)}k and the clock error of the internal clock of the rover is denoted by dtk, both of which are to be accurately estimated by the PPP technique, the following system of observation equations for the set of n GNSS satellites in view of the rover is obtained:Lci+cdti−(ρ0)ki=−({circumflex over (ρ)}0)ki·[{right arrow over (r)}k−{right arrow over (r)}0,k]+cdtk+Mki·δTk+(Bc)ki+λnwk+ε  (Eq. 3)Pci+cdti−(ρ0)ki=−({circumflex over (ρ)}0)ki·[{right arrow over (r)}k−{right arrow over (r)}0,k]+cdtk+Mki·δTk+ε′  (Eq. 4)
These ionospheric-free observation equations relate the observables obtained at step S1204 to the actual distances of the rover to the GNSS transmitters aboard the GNSS satellites, taking into account instrumental and observational errors. Therein, c is the speed of light in a vacuum, ρ0 is the approximated modeled range between the rover and the respective (i-th) satellite, and {circumflex over (ρ)}0 is the corresponding unit length vector along the direction pointing from the approximate position {right arrow over (r)}0,k of the rover to the respective (i-th) satellite. Further, M and δT are, respectively, the tropospheric mapping and residual vertical delay, w is the un-modeled user windup and wavelength λn can be obtained from the carrier frequencies f1 and f2 via
      λ    n    =            c                        f          1                +                  f          2                      .  Finally, ε and ε′ respectively represent the phase and code measurement errors associated to thermal noise and multipath, and Bc represents the ionospheric-free carrier-phase ambiguity. The expression “ambiguity” refers to an indeterminable part of the difference between the carrier phase of a GNSS signal measured at the GNSS receiver of the rover at reception time of the signal and the carrier phase measured at the respective GNSS transmitter at transmission time of the signal. For instance, for a given carrier with frequency fX, the ambiguity BXi is comprised of an integer part λXNX corresponding to an unknown number of full cycles, a first fractional part δBXi corresponding to an instrumental bias of the GNSS transmitter of the respective satellite (satellite phase bias), and a second fractional part δBX,k corresponding to an instrumental bias of the GNSS receiver of the rover (receiver phase bias). Therein, the satellite phase bias indicates a phase offset between the signal generated at the GNSS transmitter and a reference signal based on a reference time frame, and the receiver phase bias indicates a phase offset between a reference signal generated internally in the GNSS receiver and the reference signal based on the reference time frame.
In step S1205, the above system of 2n observation equations is solved by applying a so-called Kalman filter to the system of observation equations. The timing at which signals from all GNSS transmitters of the n GNSS transmitters aboard the satellites in view of the rover are simultaneously received by the GNSS receiver of the rover and are processed is referred to as an “epoch.” For each epoch, the above system of 2n observation equations can be typically solved if the number of GNSS satellites in view of the rover is equal to or exceeds n=4. By taking into account solutions from previous epochs, the accuracy of the solutions can be successively increased. In applying the Kalman filter, the rover position {right arrow over (r)}k and the user clock dtk can be treated as white noise, the residual (“wet”) delay δT and the user windup w can be treated as random walk processes, and the ionospheric-free carrier-phase ambiguity Bc is estimated as a random variable (“constant parameter”), with the exception of the occurrence of cycle-slip events. In such a case, Bc is treated as a white noise random process.
The main drawback of the above-described PPP technique is the large convergence time required to obtain a good estimation of the ionospheric-free ambiguity Bc, where Bc is to be understood as a shorthand for the set of ambiguities Bci, and correspondingly for the rover position {right arrow over (r)}k. The convergence time can last the best part of one hour—or more—before a high accuracy for the rover position at the level of 1 to 2 decimeters can be obtained.
These problems of the PPP technique limit the application of this technique to a position determination, for which the convergence time is not an issue, such as position determination for slow-moving objects, watercraft or stationary objects. The PPP technique accordingly is not applicable for determining the position of most surface vehicles, such as passenger cars, or most aircraft, such as planes. Furthermore, because of the long convergence time of the PPP technique, a continuous unobstructed view between the rover and the respective GNSS satellites is essential for the reliable operation of the PPP technique. If the tracking of individual or all satellites is lost on time scales shorter than or on the order of the convergence time, no reliable position determination might be possible at all. Therefore, the PPP technique is also not deployable for navigation in densely built areas, such as cities or in other areas in which no continuous unobstructed view between the rover and the respective GNSS satellites is present.